I'm trying to prove the following statement:
$$\forall_{a,b\in\Bbb{N^{+}}}\gcd(n^a-1,n^b-1)=n^{\gcd(a,b)}-1$$
As for now I managed to prove that $n^{\gcd(a,b)}-1$ divdes $n^a-1$ and $n^b-1$:
Without ...

Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following ...

Using Induction proof makes sense to me and know how to do, but I am having a problem in using a direct proof for practice problem that was given to us.
The problem is:
For all natural numbers $n$, ...

Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$.
This is what I have done so far:
Let $d = \gcd(a,b)$. Then $d=ax+by$ for some $x,y$. Then $d^2 =(ax+by)^2 = a^2x^2 + 2axby+b^2y^2$.
I am trying to create a ...

The Diophantine equation that I have to solve is:
$$343x^2-27y^2=1$$
This question has already been posted by other user but it has not received an answer.
I proved to solve it.
This is my attempt:
...

Im trying to solve this problem but I do not understand what the question is asking:
Let $n\ge 2$ be an integer and $ p_n(x) $ be the polynomial: $$ p_n(x) = (x-1)+(x-2)+\cdots+(x-n) $$
What is the ...

Notation: by the $d$'th diagonal of an $n \times n$ matrix $A$ I will denote the diagonal parallel to the main diagonal that starts in row 1, column $d$. I will extend this definition in the obvious ...

My first thought was that it would be $2^n$ itself, for example, if you concatenate $4$ and $2$ to get $42$, that's divisible by $2$ but not by $4$.
But whit $2^9 = 512$, you can concatenate $16$ and ...

This statement was given in my number theory textbook when analyzing quadratic fields, and I am not seeing how to prove it. $m$ is a squarefree (not divisible by the square of any number) integer and ...

Concerning a numbers’ digits we know some necessary conditions on them for the number to be prime, besides the last digit having to be odd (except for prime 2). For example in decimal representation ...

prove that if $d$ divides $n$ then prove that
fibonacci of $d$ divides fibonacci of $n$.
i have tried to write $F(n)$ as a multiple of $F(d)$
using the fact that $n = ad$ for some natural $a$ but got ...

Let $a_{1},\dots,a_{n}$ be integers. Show that there exist integers $k$ and $r$ such that the sum $$a_{k}+a_{k+1}+\dots+a_{k+r}$$ is divisible by $n$.
I am unable to find the necessary way to solve ...

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...